If your house isn’t too far from the lake, then you can easily get a very precise reading. All it requires is water, a stick, and clear tubing.
Attach the tubing to the stick, and mark it off in increments as precise as you wish. Leave the loose end of the tubing at the bottom. Attach enough tubing (or hose, or whatever) to reach the house. Have someone stand at the edge of the lake, holding the bottom (the “0” point) of the stick at precisely water level. Now, fill the whole thing with water, and lay the end by the house down on the ground. Excess water will run out.
Back at the lake end, the water level in the tube attached to the stick will show precisely how high above lake level your house is.
This gets practically much more difficult as the distance increased and the height difference. Like if the difference is 20 ft, how do you elevate the tube 20 ft at the lake?
You could do this in steps however, perhaps with 100 ft of tube, find that difference, then move it the next 100 ft, ect.
I believe I’d spend the money for a survey. As others have said, you don’t need to know the exact elevation, but the difference between the top of dam and the house finished floor.
Be aware that in catastrophic floods it’s possible for water to be flowing at some depth over the top of the dam. Sometimes that would be a foot or two, but it is possible for that depth to be several feet. Is there someone you can talk to about the design of the lake, for example what storm it’s supposed to pass without overtopping?
He hasn’t said how far the lake is from the house. If it’s 100 feet, this works well. If it’s 1,000 feet, that’s a lot of tubing. If it’s 47 miles, we have another issue entirely.
As for the 20-foot difference, you just use a 20-foot stick. Granted, you have to either have good eyes or climb up on something to read it, but it works.